Introduction to Open Data Science course, my github repository: https://github.com/peempe/IODS-project
Learning2014 dataset includes information about students who attended the basic statistics course at the end of year 2014. The dataset includes 7 variables and 166 observations. The variables are gender, age, attitude towards statistics, points in the final exam and variables addressing deep, surface, and strategic learning.
More information about the dataset can be found here.
First, we need to get the dataset to R. I had some problems with my Mac (even though I double checked it like thousand times in the “Data wrangling” part), so I decided to download it via link.
setwd("~/Documents/GitHub/IODS-project/data")
learning2014 <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/learning2014.txt",
sep = ",", header = T)
Second, we need to have an access to ggplot2 and GGally-packages. Remember to install them first!
library(ggplot2)
library(GGally)
Next, let’s see how the data looks like.
plot1 <- ggpairs(learning2014, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
plot1
Here are some main points about the output:
Most of the participants are women in their twenties.
The highest correlation is between attitude and point, addressing that a good attitude towards statistics and/or the course is associated with higher points in the final exam.
The lowest correlation is between deep learning strategy and point, indicating that they have nothing to do with each others.
Now we could examine a bit do gender, age, and attitude predict points in the final exam.
points_model <- lm(points ~ gender + age + attitude, data = learning2014)
summary(points_model)
##
## Call:
## lm(formula = points ~ gender + age + attitude, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.4590 -3.3221 0.2186 4.0247 10.4632
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.42910 2.29043 5.863 2.48e-08 ***
## genderM -0.33054 0.91934 -0.360 0.720
## age -0.07586 0.05367 -1.414 0.159
## attitude 3.60657 0.59322 6.080 8.34e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.315 on 162 degrees of freedom
## Multiple R-squared: 0.2018, Adjusted R-squared: 0.187
## F-statistic: 13.65 on 3 and 162 DF, p-value: 5.536e-08
In the model above, only attitude is statistically significant variable with p-value < .001. The model shows that better attitude towards statistics/course precits higher points in the exam. Because gender and age are not significant variables, we can remove them from the model and run the model again only with attitude-variable.
points_att <- lm(points ~ attitude, data = learning2014)
summary(points_att)
##
## Call:
## lm(formula = points ~ attitude, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.6372 1.8303 6.358 1.95e-09 ***
## attitude 3.5255 0.5674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09
The attitude is still highly significant and it explains 19 % of the value of points (R-square). Note, that the R-square value did not change much when the two other variables were removed from the model.
Last, after fitting the model and intepreting the results, we should check the diagnostic plots. The following commands do the trick:
points_att <- lm(points ~ attitude, data = learning2014)
plot(points_att, which = c(1, 2, 5), par(mfrow = c(2,2)))
All plots shows that the residuals are nicely spread on both sides of 0 (Residuals vs Fitted values), follow the line well and are (quite) normally distributed (Normal QQ-plot), and the few outliers do not influence the analysis too much (Residuals vs Leverage). The plots indicate that our model fit well to the data and the association between attitude and points is linear (as it should be when using linear regression).
alc <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/alc.txt",
sep = ",", header = T)
View(alc)
colnames(alc)
## [1] "school" "sex" "age" "address" "famsize"
## [6] "Pstatus" "Medu" "Fedu" "Mjob" "Fjob"
## [11] "reason" "nursery" "internet" "guardian" "traveltime"
## [16] "studytime" "failures" "schoolsup" "famsup" "paid"
## [21] "activities" "higher" "romantic" "famrel" "freetime"
## [26] "goout" "Dalc" "Walc" "health" "absences"
## [31] "G1" "G2" "G3" "alc_use" "high_use"
The dataset is jointed from two different datasets and it includes information about students’ alcohol consumpiton. The infromation is gathered from Portuguese and Math classes from two Portuguese schools
More information about the dataset can be found here.
The aim of this week is to analyze the relationship of high/low alcohol consumption (AC) and following variables: mother’s education (Medu), quality of family realtionships (famrel), current health status (health), and number of school absences (absences).
My hypotheses are: 1. Mother’s low education is associated with higher AC. 2. Poor quality of family relationships is associated with higher AC. 3. Poor health status is associated with higher AC. 4. Higher number of school absences is associated with higher AC.
library(ggplot2)
Mother’s education
Medutable <- table(alc$Medu,alc$high_use)
Medutable
##
## FALSE TRUE
## 0 1 2
## 1 33 18
## 2 80 18
## 3 59 36
## 4 97 38
Quality of family relationships
FamRtable <- table(alc$famrel,alc$high_use)
FamRtable
##
## FALSE TRUE
## 1 7 2
## 2 9 9
## 3 40 26
## 4 131 52
## 5 83 23
Current health status
Healthtable <- table(alc$health,alc$high_use)
Healthtable
##
## FALSE TRUE
## 1 35 11
## 2 29 14
## 3 63 20
## 4 47 17
## 5 96 50
Number of school absences
g1 <- ggplot(alc, aes(x = high_use, y = absences, col = sex))
g1 + geom_boxplot() + ggtitle("School absences by alcohol consumption")
library(magrittr)
m <- glm(high_use ~ Medu + famrel + health + absences, data = alc, family = "binomial")
summary(m)
##
## Call:
## glm(formula = high_use ~ Medu + famrel + health + absences, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.8333 -0.8113 -0.6894 1.1883 1.9684
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.50040 0.63728 -0.785 0.432326
## Medu -0.05114 0.10800 -0.474 0.635825
## famrel -0.29326 0.12388 -2.367 0.017918 *
## health 0.14367 0.08676 1.656 0.097742 .
## absences 0.06806 0.01799 3.783 0.000155 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 462.21 on 381 degrees of freedom
## Residual deviance: 435.72 on 377 degrees of freedom
## AIC: 445.72
##
## Number of Fisher Scoring iterations: 4
# computing odds ratios (OR)
OR <- coef(m) %>% exp
# computing confidence intervals (CI)
CI <- confint(m) %>% exp
## Waiting for profiling to be done...
# printing out the odds ratios with their confidence intervals
cbind(OR, CI)
## OR 2.5 % 97.5 %
## (Intercept) 0.6062858 0.1712953 2.0992320
## Medu 0.9501429 0.7689690 1.1754068
## famrel 0.7458269 0.5839293 0.9507828
## health 1.1545005 0.9762890 1.3729075
## absences 1.0704246 1.0353696 1.1107908
library(dplyr); library(ggplot2)
##
## Attaching package: 'dplyr'
## The following object is masked from 'package:GGally':
##
## nasa
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
# predicting the probability of high_use
probabilities <- predict(m, type = "response")
# adding the predicted probabilities to 'alc'
alc <- mutate(alc, probability = probabilities)
# using the probabilities to make a prediction of high_use
alc <- mutate(alc, prediction = probability > 0.5)
# seeing the last ten original classes, predicted probabilities, and class predictions
select(alc, failures, absences, sex, high_use, probability, prediction) %>% tail(10)
## failures absences sex high_use probability prediction
## 373 1 0 M FALSE 0.2677134 FALSE
## 374 1 14 M TRUE 0.3127047 FALSE
## 375 0 2 F FALSE 0.2045304 FALSE
## 376 0 7 F FALSE 0.3355262 FALSE
## 377 1 0 F FALSE 0.2057626 FALSE
## 378 0 0 F FALSE 0.1566707 FALSE
## 379 1 0 F FALSE 0.4684247 FALSE
## 380 1 0 F FALSE 0.4684247 FALSE
## 381 0 3 M TRUE 0.3210711 FALSE
## 382 0 0 M TRUE 0.2481431 FALSE
# tabulating the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)
## prediction
## high_use FALSE TRUE
## FALSE 258 12
## TRUE 96 16
# initialize a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
# define the geom as points and draw the plot
g + geom_point()
# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table() %>% addmargins()
## prediction
## high_use FALSE TRUE Sum
## FALSE 0.67539267 0.03141361 0.70680628
## TRUE 0.25130890 0.04188482 0.29319372
## Sum 0.92670157 0.07329843 1.00000000
# define a loss function (mean prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.2827225
# 10-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
# average number of wrong predictions in the cross validation
cv$delta[1]
## [1] 0.2827225
Sorry, I didn’t have time to finish the exercise.
The dataset is an example dataset in package MASS. It includes information of housing values in suburbs of Boston
More information about the dataset can be found here.
library(magrittr)
library(MASS)
##
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
##
## select
# Load the data
data("Boston")
# Explore the dataset
str(Boston) # 506 rows/obervations, 14 columns/variables
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
# Calculate the correlation matrix and round it
cor_matrix<-cor(Boston) %>% round(digits = 2)
# Print the correlation matrix
cor_matrix
## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
# Visualize the correlation matrix
library(corrplot)
## corrplot 0.84 loaded
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)
The correlation matrix gives us a nice overview of the relationships between the variables. By using colors to indicate the direction and strength of the correlation, we can see the relevant information with quick look. The matrix shows that the strongest negative correlation is between variables “rad” and “tax”. This indicates that higher index of accessibility to radial highways is associated with lower full-value property-tax rate per $10,000. The strong positive correlation can be found e.g. between variables “age” and “dis” addressing that higher proportion of owner-occupied units built prior to 1940 is associated with higher weighted mean of distances to five Boston employment centres.
# Center and standardize variables
boston_scaled <- scale(Boston)
# Summaries of the scaled variables
summary(boston_scaled)
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865
# Class of the boston_scaled object
class(boston_scaled)
## [1] "matrix"
# Change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
Scaling of the dataset standardize the variables so that the mean is 0 and standard deviation is 1.
# Summary of the scaled crime rate
summary(boston_scaled$crim)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110
# Create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins
## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610
# Create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low", "med_low", "med_high", "high"))
# Look at the table of the new factor crime
table(crime)
## crime
## low med_low med_high high
## 127 126 126 127
# Remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# Add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# Number of rows in the Boston dataset
n <- nrow(boston_scaled)
# Choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# Create train set
train <- boston_scaled[ind,]
# Create test set
test <- boston_scaled[-ind,]
# Linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# Print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2376238 0.2574257 0.2524752 0.2524752
##
## Group means:
## zn indus chas nox rm
## low 0.90649007 -0.9020745 -0.149294685 -0.8711616 0.384308674
## med_low -0.08877141 -0.2744939 -0.083045403 -0.5604705 -0.156830901
## med_high -0.38467135 0.2387383 0.152260173 0.4273526 -0.001600564
## high -0.48724019 1.0149946 -0.002135914 1.0192547 -0.454166225
## age dis rad tax ptratio
## low -0.8836172 0.8892255 -0.6756133 -0.7543941 -0.40335515
## med_low -0.3109464 0.3610218 -0.5456746 -0.4429328 -0.03897546
## med_high 0.4196039 -0.3895358 -0.4357864 -0.3093046 -0.34943789
## high 0.7795600 -0.8479649 1.6596029 1.5294129 0.80577843
## black lstat medv
## low 0.37092984 -0.72649690 0.50633079
## med_low 0.31117058 -0.12155941 -0.04883629
## med_high 0.07705465 0.03076319 0.13458388
## high -0.80492052 0.86835219 -0.64102258
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.122775006 0.651207006 -0.98213652
## indus 0.081839720 -0.280515418 0.20703380
## chas 0.014865410 -0.005224141 0.10443371
## nox 0.227296558 -0.827318076 -1.31528961
## rm 0.052458662 -0.029663220 -0.08497085
## age 0.253118465 -0.352184210 -0.02424640
## dis -0.101549847 -0.278106063 0.17861821
## rad 3.705305225 0.836121608 -0.32931968
## tax -0.002452226 0.069321385 0.77794078
## ptratio 0.156282259 0.050665799 -0.20745654
## black -0.125396447 0.026608391 0.10643251
## lstat 0.095205149 -0.122827397 0.27384024
## medv -0.014018813 -0.343981230 -0.28716328
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9586 0.0313 0.0102
# The function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# Target classes as numeric
classes <- as.numeric(train$crime)
# Plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 3)
# Save the correct classes from test data
correct_classes <- test$crime
# Remove the crime variable from test data
test <- dplyr::select(test, -crime)
# Predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# Cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
## predicted
## correct low med_low med_high high
## low 20 10 1 0
## med_low 2 17 3 0
## med_high 0 7 15 2
## high 0 0 1 24
LDA model works quite well. As assumed, the contiguous categories are harder to discriminate. However, the model does not mix extremity categories themselves at all.
# Reload the data and standardize it
data("Boston")
boston_scaled <- scale(Boston)
# Euclidean distance matrix
dist_eu <- dist(Boston)
# Look at the summary of the distances
summary(dist_eu)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.624 170.539 226.315 371.950 626.047
# Manhattan distance matrix
dist_man <- dist(Boston, method = "manhattan")
# Look at the summary of the distances
summary(dist_man)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.016 149.145 279.505 342.899 509.707 1198.265
# K-means clustering
km <-kmeans(Boston, centers = 3)
# Plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
set.seed(123)
# Determine the number of clusters
k_max <- 10
# Calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# Visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
As can be seen from the total WCSS picture, the value of total WCSS radically changes between 1 and 2,5. It indicates that the optimal number of clusters could be 2. Let’s run the algorithm again.
# K-means clustering
km <-kmeans(Boston, centers = 2)
# Plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
Now the colors red and black visualize the two clusters.
The dataset is a combination of two datasets; Human Development and Gender Inequality. After some “data wrangling”, the dataset looks like this. The wrangling steps can be seen from Rscript-file in my GitHub.
I decided to download the data from the link because somehow the mutation from string values to numeric did not saved in to the human data and I didn’t have time to figure out what was the problem.
human2 <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt",
sep = ",", header = T)
str(human2) # 155 observations and 8 variables
## 'data.frame': 155 obs. of 8 variables:
## $ Edu2.FM : num 1.007 0.997 0.983 0.989 0.969 ...
## $ Labo.FM : num 0.891 0.819 0.825 0.884 0.829 ...
## $ Edu.Exp : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ Life.Exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ Mat.Mor : int 4 6 6 5 6 7 9 28 11 8 ...
## $ Ado.Birth: num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Parli.F : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
summary(human2)
## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI Mat.Mor Ado.Birth Parli.F
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50
Let’s investigate the association between variables with a correlation matrix:
library("dplyr")
library("stringr")
library("MASS")
library("tidyr")
##
## Attaching package: 'tidyr'
## The following object is masked from 'package:magrittr':
##
## extract
library("corrplot")
library("GGally")
ggpairs(human2)
cor(human2) %>% corrplot()
pca_human <- prcomp(human2) # PCA
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("grey40", "deeppink2")) #Biplot
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
s <- summary(pca_human)
s
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000
## PC8
## Standard deviation 0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion 1.0000
pca_pr <- round(100*s$importance[2, ], digits = 1)
pca_pr
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## 100 0 0 0 0 0 0 0
Not really working with unstandardized data. Let’s try after standardation:
human2_std <- scale(human2)
summary(human2_std)
## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## GNI Mat.Mor Ado.Birth Parli.F
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850
pca_human2 <- prcomp(human2_std) # PCA
s2 <- summary(pca_human2)
s2
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
## PC7 PC8
## Standard deviation 0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion 0.98702 1.00000
pca_pr2 <- round(100*s2$importance[2, ], digits = 1)
pca_pr2
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## 53.6 16.2 9.6 7.6 5.5 3.6 2.6 1.3
pc_lab <- paste0(names(pca_pr2), " (", pca_pr2, "%)")
biplot(pca_human2, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2]) # Biplot
Now the biplot looks much better.
This data cleary need standardazion to be analyzed (and especially interpret) with PCA. The unstandardized PCA accuse that the first principal component (PC) explains 100 % of the variance and so cannot separete the effect of component. After standardazion, however, the first PCA explain 53,6 % of the variance and the second one 16,2 %. …Sorry, I didn’t have time to finish the intepretation.
First, a quick overview of the dataset:
library("FactoMineR")
data("tea")
str(tea)
## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
dim(tea) # 300 observations, 36 variables
## [1] 300 36
Second, MCA (with only few variables):
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- dplyr::select(tea, one_of(keep_columns))
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() +
theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped
mca <- MCA(tea_time, graph = FALSE)
summary(mca)
##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |
plot(mca, invisible=c("ind"), habillage = "quali")
Sorry, I ran out of time and couldn’t intepret the results. :(
library(dplyr)
library(tidyr)
First, two variables need to be factored again.
RATS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", header = TRUE, sep = '\t')
RATS$ID <- factor(RATS$ID)
RATS$Group <- factor(RATS$Group)
glimpse(RATS)
## Observations: 16
## Variables: 13
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1 <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8 <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15 <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22 <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29 <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36 <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43 <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44 <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50 <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57 <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64 <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...
RATSL <- RATS %>%
gather(key = WD, value = Weight, -ID, -Group) %>%
mutate(Time = as.integer(substr(WD,3,4)))
glimpse(RATSL)
## Observations: 176
## Variables: 5
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1",...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...
Let’s begin with plotting the data to identify the observations in each group (but ignoring the longitudinal nature of the data at this point).
library(ggplot2)
ggplot(RATSL, aes(x = Time, y = Weight, group = ID)) +
geom_line(aes(linetype = Group)) +
scale_x_continuous(name = "Time (days)", breaks = seq(0, 60, 10)) +
scale_y_continuous(name = "Weight (grams)") +
theme(legend.position = "top")
As we can see from the plot, group 1 cleary differs from two other groups. Rats in group 1 weigh less and their weight does not rise during the days compared to two other groups.
Next, we can build a linear model to examine does group and time predict weight as the plot suggest. We cannot use basic linear regression model because it assumes that the observations are independent and in this case they are not as we examine the weights of same rats. Let’s fit random intercept model to study if group and time predict the rat’s weight.
library(lme4)
## Loading required package: Matrix
##
## Attaching package: 'Matrix'
## The following object is masked from 'package:tidyr':
##
## expand
RATS_ref <- lmer(Weight ~ Time + Group + (1 | ID), data = RATSL, REML = FALSE)
RATS_ref
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: Weight ~ Time + Group + (1 | ID)
## Data: RATSL
## AIC BIC logLik deviance df.resid
## 1333.1639 1352.1868 -660.5819 1321.1639 170
## Random effects:
## Groups Name Std.Dev.
## ID (Intercept) 32.953
## Residual 8.151
## Number of obs: 176, groups: ID, 16
## Fixed Effects:
## (Intercept) Time Group2 Group3
## 244.0689 0.5857 220.9886 262.0795
After modelling we note that standard deviation is quite big suggesting large variablility between rats. Let’s take one step foward and see how the random slope model works.
Fitting a random intercept and random slope model allows the linear regression fits for each individual to differ in intercept but also in slope. This way it is possible to account for the individual differences in the rats’ growth profiles, but also the effect of time.
RATS_ref1 <- lmer(Weight ~ Time + Group + (Time | ID), data = RATSL, REML = FALSE)
RATS_ref1
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: Weight ~ Time + Group + (Time | ID)
## Data: RATSL
## AIC BIC logLik deviance df.resid
## 1194.2189 1219.5828 -589.1095 1178.2189 168
## Random effects:
## Groups Name Std.Dev. Corr
## ID (Intercept) 33.7718
## Time 0.3349 -0.22
## Residual 4.4436
## Number of obs: 176, groups: ID, 16
## Fixed Effects:
## (Intercept) Time Group2 Group3
## 246.4573 0.5857 214.5874 258.9273
The two models seem not to be very different. We can also perform an ANOVA test to compare the models.
anova(RATS_ref1, RATS_ref)
## Data: RATSL
## Models:
## RATS_ref: Weight ~ Time + Group + (1 | ID)
## RATS_ref1: Weight ~ Time + Group + (Time | ID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## RATS_ref 6 1333.2 1352.2 -660.58 1321.2
## RATS_ref1 8 1194.2 1219.6 -589.11 1178.2 142.94 2 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The interpretation of the test can be done by looking the values of chi-squared statistics and p-value. Lower value address better fit against the comparison model: the random slope model seems to fit better.
Finally, we can fit a random intercept and slope model that allows for a group × time interaction.
RATS_ref2 <- lmer(Weight ~ Time * Group + (Time | ID), data = RATSL, REML = FALSE)
summary(RATS_ref2)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: Weight ~ Time * Group + (Time | ID)
## Data: RATSL
##
## AIC BIC logLik deviance df.resid
## 1185.9 1217.6 -582.9 1165.9 166
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.2669 -0.4249 0.0726 0.6034 2.7513
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## ID (Intercept) 1.107e+03 33.2763
## Time 4.925e-02 0.2219 -0.15
## Residual 1.975e+01 4.4436
## Number of obs: 176, groups: ID, 16
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 251.65165 11.80279 21.321
## Time 0.35964 0.08215 4.378
## Group2 200.66549 20.44303 9.816
## Group3 252.07168 20.44303 12.330
## Time:Group2 0.60584 0.14229 4.258
## Time:Group3 0.29834 0.14229 2.097
##
## Correlation of Fixed Effects:
## (Intr) Time Group2 Group3 Tm:Gr2
## Time -0.160
## Group2 -0.577 0.092
## Group3 -0.577 0.092 0.333
## Time:Group2 0.092 -0.577 -0.160 -0.053
## Time:Group3 0.092 -0.577 -0.053 -0.160 0.333
anova(RATS_ref2, RATS_ref1)
## Data: RATSL
## Models:
## RATS_ref1: Weight ~ Time + Group + (Time | ID)
## RATS_ref2: Weight ~ Time * Group + (Time | ID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## RATS_ref1 8 1194.2 1219.6 -589.11 1178.2
## RATS_ref2 10 1185.9 1217.6 -582.93 1165.9 12.361 2 0.00207 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The model with interaction fits even better than the random slope model without it. We should also check the plot with observed values.
Fitted <- fitted(RATS_ref2)
RATSL <- RATSL %>%
mutate(Fitted)
ggplot(RATSL, aes(x = Time, y = Fitted, group = ID)) +
geom_line(aes(linetype = Group)) +
scale_x_continuous(name = "Time (days)", breaks = seq(0, 60, 20)) +
scale_y_continuous(name = "Fitted weight (grams)") +
theme(legend.position = "top")
First, two variables need to be factored again.
BPRS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep =" ", header = T)
BPRS$treatment <- factor(BPRS$treatment)
BPRS$subject <- factor(BPRS$subject)
glimpse(BPRS)
## Observations: 40
## Variables: 11
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ week0 <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week1 <int> 36, 68, 55, 77, 75, 43, 61, 36, 43, 51, 34, 52, 32, ...
## $ week2 <int> 36, 61, 41, 49, 72, 41, 47, 38, 39, 51, 34, 49, 36, ...
## $ week3 <int> 43, 55, 38, 54, 65, 38, 30, 38, 35, 55, 41, 54, 31, ...
## $ week4 <int> 41, 43, 43, 56, 50, 36, 27, 31, 28, 53, 36, 48, 25, ...
## $ week5 <int> 40, 34, 28, 50, 39, 29, 40, 26, 22, 43, 36, 43, 25, ...
## $ week6 <int> 38, 28, 29, 47, 32, 33, 30, 26, 20, 43, 38, 37, 21, ...
## $ week7 <int> 47, 28, 25, 42, 38, 27, 31, 25, 23, 39, 36, 36, 19, ...
## $ week8 <int> 51, 28, 24, 46, 32, 25, 31, 24, 21, 32, 36, 31, 22, ...
BPRSL <- BPRS %>% gather(key = weeks, value = bprs, -treatment, -subject)
BPRSL <- BPRSL %>% mutate(week = as.integer(substr(weeks,5,5)))
glimpse(BPRSL)
## Observations: 360
## Variables: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
Let’s begin by plotting the BPRS values for all 40 men, differentiating between the treatment groups into which the men have been randomized.
ggplot(BPRSL, aes(x = week, y = bprs, linetype = subject)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(BPRSL$bprs), max(BPRSL$bprs)))
At this point the plot mostly shows that even though there is a lot of variability between individuals, there seems to be less psychotic symptoms in group 1 after 8 weeks.
Next we should do some analysis to examine if this hypothesis is correct. First we can standardize the values of BPRS.
BPRSL <- BPRSL %>%
group_by(week) %>%
mutate(stdbprs = (bprs - mean(bprs)) / sd(bprs)) %>%
ungroup()
glimpse(BPRSL)
## Observations: 360
## Variables: 6
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
## $ stdbprs <dbl> -0.4245908, 0.7076513, 0.4245908, 0.4953559, 1.69836...
Let’s plot the data again, now with the standardized values.
ggplot(BPRSL, aes(x = week, y = stdbprs, linetype = subject)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_y_continuous(name = "standardized bprs")
After standardizing the values, the plot shows that the variability is still large but now in group 1 psychotic symptoms do not seem to reduce.
Next we can try produce a graph showing average profiles for each treatment group along with some indication of the variation of the observations at each time point.
n <- BPRSL$week %>% unique() %>% length()
BPRSS <- BPRSL %>%
group_by(treatment, week) %>%
summarise( mean = mean(bprs), se = sd(bprs) / sqrt(n) ) %>%
ungroup()
glimpse(BPRSS)
## Observations: 18
## Variables: 4
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2
## $ week <int> 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8
## $ mean <dbl> 47.00, 46.80, 43.55, 40.90, 36.60, 32.70, 29.70, 29....
## $ se <dbl> 4.534468, 5.173708, 4.003617, 3.744626, 3.259534, 2....
# Plot the mean profiles
ggplot(BPRSS, aes(x = week, y = mean, linetype = treatment, shape = treatment)) +
geom_line() +
scale_linetype_manual(values = c(1,2)) +
geom_point(size=3) +
scale_shape_manual(values = c(1,2)) +
geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.8)) +
scale_y_continuous(name = "mean(bprs) +/- se(bprs)")
Okay, the two groups seems to overlap quite much. Let’s find out if there are an outlier which could explain this situation.
BPRSL8S <- BPRSL %>%
filter(week > 0) %>%
group_by(treatment, subject) %>%
summarise( mean=mean(bprs) ) %>%
ungroup()
glimpse(BPRSL8S)
## Observations: 40
## Variables: 3
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ mean <dbl> 41.500, 43.125, 35.375, 52.625, 50.375, 34.000, 37.1...
# Draw a boxplot of the mean versus treatment
ggplot(BPRSL8S, aes(x = treatment, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(bprs), weeks 1-8")
Hahaa, there is one outlier in group 2. Let’s filter that one out and see the situation once again.
BPRSL8S1 <- BPRSL8S %>% filter(mean < 70)
glimpse(BPRSL8S1)
## Observations: 39
## Variables: 3
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ mean <dbl> 41.500, 43.125, 35.375, 52.625, 50.375, 34.000, 37.1...
ggplot(BPRSL8S1, aes(x = treatment, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(bprs), weeks 1-8")
Now it looks like that there are less psychotic symptoms in group 2. Next we should do some analysis to be more sure about it.
t.test(mean ~ treatment, data = BPRSL8S1, var.equal = TRUE)
##
## Two Sample t-test
##
## data: mean by treatment
## t = 0.52095, df = 37, p-value = 0.6055
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -4.232480 7.162085
## sample estimates:
## mean in group 1 mean in group 2
## 36.16875 34.70395
BPRSL8S2 <- BPRSL8S %>% mutate(baseline = BPRS$week0)
fit <- lm(mean ~ baseline + treatment, data = BPRSL8S2)
anova(fit)
## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 1868.07 1868.07 30.1437 3.077e-06 ***
## treatment 1 3.45 3.45 0.0557 0.8148
## Residuals 37 2292.97 61.97
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Yep, the t-test confirms that the BPRS score is lower in group 2 after 8 weeks.